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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2021

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Volume 2021 |Article ID 1891749 | https://doi.org/10.1155/2021/1891749

Zhuoran Fan, Jilong Xu, Yuchen Li, "Multiobjective Programming Strategy of Small- and Medium-Sized Microenterprise Credit Based on Random Factors", Journal of Function Spaces, vol. 2021, Article ID 1891749, 10 pages, 2021. https://doi.org/10.1155/2021/1891749

Multiobjective Programming Strategy of Small- and Medium-Sized Microenterprise Credit Based on Random Factors

Academic Editor: Chun Lu
Received23 Apr 2021
Accepted10 Jun 2021
Published23 Jun 2021

Abstract

In this paper, we select eight indicators from the aspects of an enterprise’s bill transaction information, namely, whether the enterprise’s loan is in breach of contract, effective invoice rate, total utilization rate of price and tax, negative invoice rate, strength of enterprise, coefficient of variation, flow efficiency of assets, and influence of upstream and downstream enterprises; then, we construct an evaluation index system. According to different industries, different categories, and the impact of random factors, we divide the types of enterprises into 10 categories. Then, we use three kinds of Poisson random numbers to carry out numerical simulation on the total price and tax of enterprises in different industries under the influence of COVID-19.

1. Background

When banks provide loans to small- and medium-sized and microenterprises (small- and medium-sized and microenterprises are abbreviated as SMMEs), they often judge whether to lend or not through credit risk assessment. Because of the lack of mortgage assets in SMMEs, the bank will make credit risk assessment based on the credit policy, influence, strength, and stability of supply and demand relationship of the enterprise, and then determine whether to lend, loan amount, interest rate and term, and other credit strategies. Some corporate banks have credit records, some have no credit records. However, in the face of the impact of sudden random factors on enterprises, how to give the credit strategy when the annual total credit is fixed.

2. The Selection of Credit Risk Quantitative Index

This paper analyzes the relevant data indicators of enterprises with credit records, takes into account the actual situation affecting the credit problems of SMMEs and refers to the advanced international standards, and selects eight quantitative indicators affecting the credit risk of enterprises according to China’s national conditions and the bank’s credit policy: (1)Whether the enterprise loan is in breach of contract is an important indicator for the bank to examine whether the enterprise can bring the money. Default is 0 and nondefault is 1. means whether th enterprise is in breach of contract. means the enterprise defaults, while means that the enterprise has not breached the contract(2)Effective invoice rate: it is equal to the ratio of the number of valid invoices to the total number of invoices. is used to denote the effective invoice rate of the th enterprise, indicates the number of valid invoices for the th enterprise, and represents the total invoice number of the th enterprise. Thus, the corresponding formula of the effective invoice rate of the th enterprise is as follows: (3)Utilization rate of total price and tax: it is equal to the ratio of the total price and tax of the valid invoice to the total price and tax of all invoices. Putting represents the utilization rate of the total price and tax of an effective invoice of the th enterprise(4)Negative rate of invoice : it is equal to the ratio of the number of invoices of the th enterprise whose value of the total invoice price and tax is “-” to the number of total invoices of the th enterprise(5): it is equal to the ratio of the difference between the total price and tax of the output and input of the th enterprise to the total price and tax of input, which indicates the strength of the enterprise. Putting represents the total price and tax of the output (sales revenue) of the th enterprise, and represents the total price and tax of the th enterprise’s input (purchased products), which uses the following corresponding formula: (6)Coefficient of variation: it indicates the stability of supply and demand relationship of enterprises. Using represents the coefficient of variation of the th enterprise. represents the total input price and tax of the th enterprise in the th month, is the total output value tax of the th enterprise in the th month, and represents the net income of the th enterprise in the th month. If , let us take directly . The corresponding formula is follows: where represents the average value of in 12 months of the th enterprise (7)Liquidity efficiency of assets: it refers to the comparative relationship between current assets and current liabilities of SMMEs in the same period, that is, the short-term solvency of SMMEs

The following table shows the asset flow data of the th enterprise in 12 months, as shown in Table 1.


Month1212

Total of input price and tax
Total of output price and tax
Net income

The net income of the previous month is transferred to the next month as part of the next month’s input, which shows the liquidity of funds. The liquidity efficiency of the th enterprise asset can be expressed as follows: where 12 represents 12 months, and refers to the number greater than 0 in of th enterprise in 12 months. In fact, is the proportion of the number of months whose value is greater than 0 to the total number of months. The larger the value indicates that the better the flow efficiency of th enterprise. (8)Influence of upstream and downstream enterprises : the influence is expressed by the maximum number of effective cooperation between the th enterprise and upstream and downstream enterprises. In order to quantitatively describe the influence of upstream and downstream enterprises, the influence function of upstream and downstream enterprises is introduced with reference to the negative exponential function of the psychological curve [1]:

where refers to the largest number of input invoice and output invoice of the th enterprise in 12 months. Following the increase of , the influence of upstream and downstream enterprises will also increase.

In the quantitative index system affecting the credit risk of SMMEs, the first to fourth indexes reflect the reputation of enterprises, the fifth index reflects the strength of enterprises, the sixth index reflects the stability of the supply and demand relationship of enterprises, the seventh index reflects the size of the credit risk of enterprises, and the eighth index reflects the influence of enterprises and upstream and downstream enterprises.

3. Comprehensive Evaluation of Credit Risk Quantitative Index System

In order to eliminate dimension and the positive and negative effects of index, in this paper, the fuzzy membership method is used to standardize the index. Let be the th index value of the th evaluation object; be the standardized value of the th index of the th evaluation object and be the number of objects to be evaluated. Then, the positive index standardization formula (6) and the negative index standardization formula (7) can be used to standardize the index [2]:

Among the 8 indicators of the quantitative index system affecting credit risk of SMMEs, the fourth indicator (negative invoice rate) and the sixth indicator (enterprise coefficient of variation) are both negative indicators, which need to be processed with the help of formula (7), while other indicators are calculated with the help of formula (6).

The following uses the entropy weight TOPSIS method to evaluate the credit risk quantitative index system of SMMEs. On the one hand, the entropy weight method is used to determine the coefficient of the credit risk quantitative index system. On the other hand, the TOPSIS method, that is, the technology of approaching the ideal solution, is used to determine the ranking of the evaluated object SMMEs. The core idea of the TOPSIS method is to define the positive ideal solution and negative ideal solution of the decision problem, and then compare and evaluate the distance between the solution and the positive ideal solution and negative ideal solution, and finally calculate the relative closeness degree between each solution and the ideal solution, and order the advantages and disadvantages of the solution.

3.1. Entropy Weight Method Being Used to Calculate the Objective Weight of Indexes

Set as the normalized value of the th indicator in the th system, where and . For a given index , the larger the difference of , the larger the comparative effect of this index has on the system, that means the more information the index contains and transmits.

The specific steps of the entropy method to determine the index weight are as follows: (i)Calculating the entropy value of 8 indicators such as effective invoice rate. Set as the entropy value of theth index, the solution process is as follows [3]:where is the characteristic proportion of the th index in the th system, and . are the sum of all system observation data of the th indicator (ii)Calculation of the coefficientof variance of the th index (iii)Determine the weight coefficients of 8 indexes

3.2. Weighting of Standardized Data

Let be the weighted value of the th index standardized data of the th SMMEs, be the normalized value of the th index observed value of the th SMMEs, and be the weight coefficient. According to the weighting method, it can be seen that

3.3. Determining the Positive and Negative Ideals of the Evaluation System

Set and as the maximum and minimum value of the th index observation data, respectively, :

It is easy to know that the positive and negative ideal solutions of the evaluation system are, respectively, and .

3.4. Calculating the Euclidean Distance between the Evaluation System and the Ideal Solution

Let be the Euclidean distance between the weighted value of the th enterprise and the positive ideal solution and be the Euclidean distance between the weighted value of the th enterprise and the negative ideal solution. Then

3.5. Calculating the Relative Closeness Evaluation Result

Set as the relative closeness of all the indexes and the ideal solution of the th enterprise, then where .

Determine the development status of the evaluated index by calculating the closeness. The greater the relative closeness , the closer the evaluated index is to the ideal solution, and the better the development status.

4. Banks’ Credit Strategies for SMMEs under Random Factors

Let be the amount of the bank’s loan to the th SMMEs and be the interest rate of the bank’s loan to the th SMMEs. Whether the bank gives loans to th SMMEs, we use the function

The production, operation, and economic benefits of enterprises may be affected by some unexpected factors, and the size of the impact is related to different industries and different types of enterprises. For example, when COVID-19 became widespread, the demand for medical services and products produced by healthcare companies increased rapidly. With the help of relevant state policies, the total credit amount of banks to such healthcare companies and health enterprises will increase. At the same time, in order to avoid the rapid transmission of COVID-19, the state often needs to cut off some transmission routes. For example, during the outbreak of COVID-19, the state issued policies to close some self-employed small- and medium-sized enterprises, so as to reduce the movement of people and avoid cross-infection caused by too many people. In this regard, banks will reduce the total amount of credit to such self-employed SMMEs to avoid credit risk.

According to different industries, different categories, and the size of the impact, we classify enterprises as follows: self-employed enterprises, trade and transportation industry, literature and art advertising industry, manufacturing industry, service industry, financial investment industry, medical and health industry, high-tech enterprises, catering industry, and other industries.

In order to visually show the impact of credit risk and possible sudden factors on each enterprise, we carry out the numerical fluctuation of the total input price tax and the total output price tax of 10 types of enterprises. According to the actual impact of COVID-19 on society, the total input price and tax and the total output price and tax of the medical and health industry should be increased, while the total input price and tax and the total output price and tax of the individual business should be reduced. The concrete method is to add random numbers (Poisson random numbers) that are divided into three types for simulation.

The first category is to increase the total input and output tax of the medical and health industry, and the total input and output tax are, respectively, and . By adding random number (0~100%), the total input price and tax and the total sales tax after the influence are, respectively and . In MATLAB software, the function is used to achieve this [4, 5].

In the second category, for self-employed enterprises, the total input price and tax and the total output price and tax of the catering industry are reduced. The original total input price and tax and the total output price and tax are, respectively, and . By adding random number (-100%~0), the total input price and tax and the total output price and tax are and . This is achieved with the help of the function.

In the third category, the influence of other industries is relatively small, and the random number (-50% ~50%) is added and fluctuates randomly, and the original total of the input price and tax and the total of the output price and tax are and , respectively. The total input price and tax and output price and tax are and . This is achieved with the help of the function .

5. Multiobjective Planning Strategy of SMME’s Credit under Random Factors

When the COVID-19 outbreak occurred, the demand for services and products provided by medical and health enterprises also increased rapidly, and the resulting enterprise profits also increased, so the ability of enterprises to repay loans increased. It is a pity that the profit of the self-employed enterprise is reduced or stagnated, and the ability to repay the loan is weakened. Due to the impact of unexpected factors, the repayment ability is weakened and the bank’s income is affected.

5.1. Determination of Objective Function

From the front, we can see that means the comprehensive evaluation score of the th enterprise out of enterprises. Let

Let the th enterprise repayment for the bank loan ratio be . Taking it here

Thus, the amount of the loan that the th enterprise can repay is . To establish the objective function

On the other hand, the smaller the bank’s lending risk, the better. indicates the unit capital risk of the th SMMEs, and represents the investment risk brought by the capital flow of the th SMMEs, and establishes an objective function for this purpose:

On the other hand, the bank loan amount should take into account the business strength of the enterprise. This paper uses the sample variance index of loan amount and total input price and tax to describe the balance of credit amount, and establishes the objective function for this purpose:

5.2. Determination of Constraints

(i)The loan limit of the established bank to the enterprise determined to be loaned is 10-100 (ten thousand), so (ii)The annual loan interest rate of the bank to the enterprise determined to lend is 4%~15%. Thus (iii)The balance of a bank’s investment in enterprises. It’s represented by . The demand for services and products provided by medical and health enterprises is increasing rapidly. Therefore, the total input price and tax of such enterprises should also be increased, and the amount of bank loans to such enterprises should be increased. When the number of self-employed enterprises decreases or stagnates, the total input value and tax should also be reduced, and the amount of bank loans to such enterprises should be reduced. The upper and lower limits of the total balance of input price and tax for medical and health input are adjusted to 0.8 and 2. Considering that an individual business cannot be given a loan completely, the upper and lower limits of the total balance of input price and tax of an individual business are adjusted to 0.3 and 1, the upper and lower limits of other industries remain at 0.5 and 1.5. Set = “medical enterprise code”; = “individual enterprise code”; and = “all other enterprise codes”. We agreed that (iv)Whether the bank loans to the enterprise and the loan amount is consistent, let be a very small positive number and be a very large positive number. The values of 1 and 0 of , respectively, indicate that the bank loans to the th enterprise and does not lend to the th enterprise. In order to ensure the consistency of bank loans to the enterprise and the loan amount, there are constraints (v)Total amount of loan. Assuming that the total amount of loan is 100 million when the bank loans to enterprises, the unit here takes 10000 yuan. We have

6. Example Checking

This paper verifies the multiobjective planning strategy of SMMEs under the influence of COVID-19 by using the related data. The original data of this paper comes from the data of competition question C for CUMCM-2020 (China University mathematical modeling competition), which can be downloaded publicly [6] (http://www.mcm.edu.cn/html_cn/node/10405905647c52abfd6377c0311632b5.html).

Firstly, the Poisson random number is considered, and with the help of the TOPSIS evaluation method, the scores and ranking comparison table of 302 enterprises before and after the introduction of random distribution are obtained [717]. The scores and ranking of the top 20 enterprises with enterprise number before and after the introduction of random distribution are shown in Table 2.


The score results of entropy weight method before introducing a random distributionThe score results of entropy weight method after introducing a random distribution
Enterprise numbersScore RankingEnterprise numbersScore Ranking

10.0347054613310.01051545258
20.0347387113220.01009714271
30.265896537730.266434252
40.29959602840.360794183
50.268823092150.3180500322
60.26631946160.0403875692
70.266693824670.01158391189
80.266329325980.0171994112
90.266816154090.01643889114
100.2661113572100.1152428472
110.2690960619110.3203013917
120.266719544120.3159082732
130.2670229931130.01160196187
140.2662758163140.1128004275
150.2669913133150.3160921931
160.2768057315160.3268668412
170.246937690170.2923549847
180.2667465742180.3171176926
190.2666004150190.3152892940
200.2458391193200.2409104158

It can be seen from Table 2 that, after the introduction of random distribution, the ranking of enterprises with enterprise labels ranging from 1 to 20 changed correspondingly—some changed greatly, while some changed less—indicating that our model has good practicability.

After the introduction of random distribution, the changes in scores and rankings.

of the top 20 enterprises among the 302 enterprises are shown in Table 3.


The score results of entropy weight method before introducing a random distributionThe score results of entropy weight method after introducing a random distribution
Enterprise numbersScore RankingEnterprise numbersScore Ranking

2060.6535376212060.745700251
300.515307012300.557323062
40.3607941832370.333588543
1070.3489208442350.314453084
920.344361555890.311441365
890.3436376861070.303524146
760.3428157572200.301107067
2200.34233996840.299596028
1220.331085459920.29288799
260.3289350210760.2912059110
620.32855488111220.2821742411
160.3268668412260.2812810112
380.3260035713380.2800868413
450.3229968214620.2781755714
1100.3216614715160.2768057315
330.3204474716450.2723488616
110.3203013917330.2712848317
530.3194912181100.2711215618
1110.3194650119110.2690960619
630.3185621320630.2689515920

As can be seen from Table 3, after the introduction of random distribution, the number of the top 20 enterprises is basically still in the top 20, indicating that our comprehensive evaluation method is relatively good and the ranking distribution is relatively stable.

From Table 4, we can see the ranking changes of enterprises in the case of occurrence of emergent factors and absence of emergent factors. It can be found that under the influence of COVID-19, the rating and ranking of enterprises in the medical and health industry have increased, indicating that under the influence of COVID-19, such enterprises have a good credit situation and a low credit risk. However, the decline in the score and ranking of self-employed enterprises indicates that under the influence of COVID-19, the credit situation of such enterprises is poor and the credit risk is high, which is in line with the actual situation. It indicates that our TOPSIS evaluation method is effective and can be better applied to the situation when random factors occur.


The score results of entropy weight method before introducing a random distributionThe score results of entropy weight method after introducing a random distribution
Enterprise numbersScore RankingEnterprise numbersScore Ranking

E195 (medical)0.26661048E195 (medical)0.31438044
E398 (medical)0.014558189E398 (medical)0.012037166
E420 (medical)0.014021227E420 (medical)0.013773184
E373 (individual)0.014911162E373 (individual)0.009712279
E124 (individual)0.034705133E124 (individual)0.010516258
E125 (individual)0.034739132E125 (individual)0.010516271

When the total annual credit of the bank is 100 million yuan, we use the data given in the attached table of question C to establish the multiobjective programming model of 302 enterprises.

The multiobjective function includes the following: , , and

The constraint conditions are as follows:

The software programming of Lingo is used to solve the above multiobjective function programming model [18]. In the three target functions, let be the scale coefficient of the th objective function, which satisfies

Set three different proportional coefficients and get different results of different loan amounts, which are analyzed in the following list (see Table 5).


Plan12345678

0.70.60.60.50.60.50.40.3
0.20.20.10.250.150.150.150.15
0.10.20.30.250.250.350.450.55
The number of different values of the loan amount799811171920

According to the analysis in Table 5, the balance of credit amount is important for banks. Finally, the eighth plan is selected to obtain the specific credit plan for 302 SMMEs, as shown in Table 6 below. The loan amount of enterprises not listed in Table 6 is 100,000 yuan.


Enterprise numberLoan amount

133880344.8
147548104
149864340.1
166532681.5
167867680.4
171150709.4
173726597.9
178836082.4
179646818.7
182425921
187402967.4
190414155.8
191735308.3
193338920
198732913.7
205313454.6
211798931.5
242184068.4
11000000
21000000
2121000000
2961000000
391000000
581000000
1001000000
1011000000
1031000000
1091000000
1241000000
1251000000
1261000000
1271000000
1281000000
1291000000
1301000000
1311000000
1321000000
1341000000
1351000000
1361000000
1371000000
1381000000
2291000000
31000000
1391000000
1401000000
1411000000
1421000000
1451000000
1481000000
1501000000
1531000000
1541000000
1551000000
1561000000
1571000000
1581000000
1591000000
1601000000
1611000000
1621000000
1631000000
1641000000
1651000000
2501000000
4100000
1691000000
1701000000
1741000000
1751000000
1761000000
1771000000
1811000000
1831000000
1841000000
1851000000
1861000000
1891000000
1961000000
1971000000
1991000000
2001000000
2011000000
2021000000
2031000000
2041000000
2581000000
5100000

It can be found that during the COVID-19 epidemic, due to the rapid increase in the demand for the services and products provided by the medical and health enterprises, the investment of the finally obtained bank in this industry also increased, the sales of the self-employed industry decreased or stagnated, and the investment of the finally obtained bank in this industry also decreased or stagnated. The investment of banks in other industries is also adjusted accordingly to maintain the survival and operation of the industry, which is more consistent with the actual situation and demonstrates the effectiveness and practicability of our model.

7. Sensitivity Analysis of the Model

The sensitivity of the model is used to analyze the sensitivity and stability of the model. In order to test the stability and effectiveness of the multiobjective programming strategy model, sensitivity analysis is carried out.

In the case of COVID-19, 3 random Poisson numbers were added to simulate the total value of the total tax and total sales tax in each industry. In order to fully demonstrate the sensitivity of the model, we changed the number of algorithm runs and record the average scores of the 302 comprehensive evaluation scores obtained by each algorithm; then, the average score of the comprehensive evaluation of the 20 algorithms was obtained and drawn. The results are shown in Figure 1.

By checking whether the average value of the comprehensive evaluation score is stable and centralized when the algorithm runs for 20 times, we can verify whether the model is stable. One can find that the average value of the comprehensive evaluation score of 20 times is relatively centralized and stable, floating in a certain range. It can be seen that our model is stable and practical.

In this paper, the TOPSIS evaluation method is firstly used to get the score and ranking comparison of 302 enterprises before and after the introduction of a random number, and the ranking results are analyzed. Then, considering that different industries each have a different ability to repay loans when affected by COVID-19, the ratio factor that can repay bank loans is introduced, and considering the floating amount of loans in different industries under national conditions and policies, the objective function and constraints of the multiobjective credit optimization model are modified, and the multiobjective credit optimization model of enterprises influenced by COVID-19 is established. When the total amount is 100 million, the corresponding credit decision is made. Finally, sensitivity analysis is carried out to test the stability and effectiveness of the multiobjective programming strategy model.

Data Availability

The original data of this paper comes from the data of competition question C for CUMCM-2020 (China University mathematical modeling competition), which can be downloaded publicly. Download from the following website: http://www.mcm.edu.cn/html_cn/node/10405905647c52abfd6377c0311632b5.html. The later data used to support the findings of this study are included within the supplementary information file(s).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of Shandong Province of China (No. ZR2020QA078) and the National Natural Science Foundation of China (No. 12005110).

Supplementary Materials

We have uploaded three attachments. Annex 1.1: raw data on 302 enterprises without credit records which comes from the data of competition question C for CUMCM-2020. Annex 2: in this paper, we get the relevant data of 302 enterprises before and after the introduction of random distribution. Annex 3: Ligon Program Source Code. (Supplementary Materials)

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